Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
LPMMC | Grenoble | 07.11.2023
Extreme Value Theory
And
LocalIZation IN RANDOM spin chains
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tossing a coin
1
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
\(L = 176 \)
HTTHTHH
THTHHT
Tossing a coin
2
HTTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTHTHTTHTHTHTHHHTHTHTHTTHHTHTHTHTHHTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (81 T / 95 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tossing a coin
HTTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTHTHTTHTHTHTHHHTHTHTHTTHHTHTHTHTHHTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (81 T / 95 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
\mathcal{P}(\ell) \sim 2^{-\ell}
\mathcal{P}(\ell_{\max}) \sim \frac{1}{L}
\Rightarrow \ell_{\max} \sim \ln L / \ln 2 \sim 7.45
2
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Extreme value theory
Athletic records
Market risks
Extreme floods
Large wildfires
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
3
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Extreme value theory
Athletic records
Market risks
Extreme floods
Large wildfires
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
3
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Condensed matter
Extreme value theory
Disordered spin chains
Athletic records
Market risks
Extreme floods
Large wildfires
Condensed matter
R. Juhász, Y,C. Lin, and F, Iglói, Phys. Rev. B 73, 224206 (2006)
N. Pancotti, M. Knap, D. A. Huse, J. I. Cirac, and M. C. Bañuls, Phys. Rev. B 97, 094206 (2018)
I. A. Kovács, T.Pető, and F.Iglói, Phys. Rev. Res. 3, 033140 (2021)
W.-H. Kao and N, B. Perkins, Phys. Rev. B 106, L100402 (2022)
...
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
3
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Extreme value theory
Disordered spin chains
Athletic records
Market risks
Extreme floods
Large wildfires
Condensed matter
R. Juhász, Y,C. Lin, and F, Iglói, Phys. Rev. B 73, 224206 (2006)
N. Pancotti, M. Knap, D. A. Huse, J. I. Cirac, and M. C. Bañuls, Phys. Rev. B 97, 094206 (2018)
I. A. Kovács, T.Pető, and F.Iglói, Phys. Rev. Res. 3, 033140 (2021)
W.-H. Kao and N, B. Perkins, Phys. Rev. B 106, L100402 (2022)
...
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
3
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Magnetization
Summary : Spin-1/2 chain in random field
4
Spin-1/2 \(W = h = \) disorder strength for random fields along \(S^z\)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
4
Spin-1/2 \(W = h = \) disorder strength for random fields along \(S^z\)
Heisenberg chain
XX chain ("many-body Anderson")
Summary : Spin-1/2 chain in random field
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
4
Spin-1/2 \(W = h = \) disorder strength for random fields along \(S^z\)
E_m
Eigenstates in the middle of the many-body spectrum
Heisenberg chain
XX chain ("many-body Anderson")
Summary : Spin-1/2 chain in random field
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Heisenberg chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Heisenberg chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Heisenberg chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
QUestions
Spin-1/2 \(W = h = \) disorder strength for random fields
6
W
Ergodic
?
MBL regime(s)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Spin-1/2 \(W = h = \) disorder strength for random fields
6
W
Ergodic
MBL regime(s)
?
Fate of isolated
quantum systems?
QUestions
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Spin-1/2 \(W = h = \) disorder strength for random fields
6
W
Ergodic
?
MBL regime(s)
QUestions
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
- Toy model : single \(\xi\)
Spin-1/2 \(W = h = \) disorder strength for random fields
- Quantitative description?
6
W
Ergodic
?
MBL regime(s)
QUestions
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
- Toy model : single \(\xi\)
Spin-1/2 \(W = h = \) disorder strength for random fields
- Consequences?
6
W
Ergodic
?
MBL regime(s)
QUestions
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
- Toy model : single \(\xi\)
- Quantitative description?
Scope
1. Spin chains in random field and localization : key points
2. Exact diagonalization
4. Quantitative analysis: extreme value theory
3. Minimal deviations in the XX chain
5. Consequences in the Heisenberg chain
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Spin chains in Random field and localization
Spin-1/2 chain in a random field
7
\mathcal{H} = \sum_{i} \frac{J}{2}\left(S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i} h_i S_i^z
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Spin-1/2 chain in a random field
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left(c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}+2 \Delta n_i n_{i+1} \right) -h_i n_i\Bigr]
Jordan-Wigner
Spinless fermions
(hardcore bosons)
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
\mathcal{H} = \sum_{i} \frac{J}{2}\left(S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i} h_i S_i^z
7
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Spin-1/2 chain in a random field
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Jump
Magnetic field
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\mathcal{P}(h_i) =
Spin-flip
On-site energy
\(-W\)
\(W\)
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
Attraction/ repulsion
Attraction/ repulsion
Ising interaction
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
7
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Anderson Localization (1D)
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H}_f = \sum_{m} \epsilon_m b_m^{\dagger}b_m
9
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\epsilon_m
1 particle
{\xi}(E, {\color{#76a5af}W})
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
\mathcal{H}_f = \sum_{m} \epsilon_m b_m^{\dagger}b_m
9
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Anderson Localization (1D)
1 particle
{\xi}(E, {\color{#76a5af}W})
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
\mathcal{H}_f = \sum_{m} \epsilon_m b_m^{\dagger}b_m
10
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Anderson Localization (1D)
\(\epsilon\)
\(\xi(\epsilon, W)\)
"Many-Body" Anderson Insulator (= XX chain)
\(L/2\) fermions
\(S_z = 0\)
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\epsilon_m
11
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\(L/2\) fermions
\(S_z = 0\)
|\Psi \rangle = | \left\{\phi_m, m \in {\color{#56B4E9}\mathrm{occ}} \right\}\rangle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\epsilon_m
E_m
"Many-Body" Anderson Insulator (= XX chain)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
11
Localization length
{\xi}(E, {\color{#76a5af}W})
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\(\epsilon\)
\(\xi(\epsilon, W)\)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
12
Localization length
{\xi}(E, {\color{#76a5af}W})
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\(\epsilon\)
\(\xi(\epsilon, W)\)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
12
Localization length
{\xi}(E, {\color{#76a5af}W})
\xi = \frac{1}{\ln\left(1+\left(\frac{W}{W_0}\right)^2 \right)}
\(\epsilon\)
\(\xi(\epsilon, W)\)
\xi \ll 1
W \gg 2
A. C. Potter, R. Vasseur, and S. A. Parameswaran, PRX 5, 031033 (2015)
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
12
Heisenberg: Effect of interactions?
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
13
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\mathcal{H} = \sum_m \epsilon_m b_m^{\dagger} b_m + \sum_{j,k,l,m} {\color{#ff9900}V_{j,k,l,m} b_j^{\dagger} b_k^{\dagger} b_l b_m}
In the Anderson basis:
Anderson
orbitals \(m\)
13
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Heisenberg: Effect of interactions?
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\mathcal{H} = \sum_m \epsilon_m b_m^{\dagger} b_m + \sum_{j,k,l,m} {\color{#ff9900}V_{j,k,l,m} b_j^{\dagger} b_k^{\dagger} b_l b_m}
In the Anderson basis:
Anderson
orbitals \(m\)
14
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
Interactions favor delocalization. Do they fully destroy localization?
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Heisenberg: Effect of interactions?
Effect of interactions? GRound state
15
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Effect of interactions? GRound state
15
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
T. Giamarchi and H. J. Schulz, EPL 3 1287 (1987); PRB 37, 325 (1988)
Z. Ristivojevic, et al PRL 109, 026402 (2012);
Doggen et al, PRB 96, 180202(R) (2017)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Effect of interactions?
16
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
What about high temperatures / high energy eigenstates?
T. Giamarchi and H. J. Schulz, EPL 3 1287 (1987); PRB 37, 325 (1988)
Z. Ristivojevic, et al PRL 109, 026402 (2012);
Doggen et al, PRB 96, 180202(R) (2017)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Effect of interactions?
16
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
T. Giamarchi and H. J. Schulz, EPL 3 1287 (1987); PRB 37, 325 (1988)
Z. Ristivojevic, et al PRL 109, 026402 (2012);
Doggen et al, PRB 96, 180202(R) (2017)
What about high temperatures / high energy eigenstates?
J. M. Deutsch , PRA. 43, 2046–2049, (1991) ,
M. Srednicki, PRE 50, 888–901, (1994)
L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016)
Do isolated quantum systems thermalize?
Thermal average
?
ETH
Time average
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Weak interactions and disorder
17
Analytical, general picture:
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
L. Fleischman, P. W. Anderson, PRB 2, 2336 (1980) \(\rightarrow\) single-particle excitations and conditions for Anderson transition
B. Altschuler, Y. Gefen, A. Kamenev, L. S. Levitov, PRL 78, 2803, (1997) \(\rightarrow\) quasi particle lifetime & localization in Fock space
P. Jacquod, D. L. Shepelyansky, PRL 79, 1837 (1997) \(\rightarrow\) Gap ratio statistics, finite systems
I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, PRL 95, 206603 (2005) \(\rightarrow\) zero conductivity at low temperature
*D. M. Basko, I. L. Aleiner, B. L. Altschuler, Annals of Physics 321, 1126 (2006) \(\rightarrow\) metal-insulator transition, localization in Fock space
I.L. Aleiner, B. L. Altshuler, G. V Shlyapnikov, Nature Physics 6, 900-904 (2010) \(\rightarrow\) weakly interacting bosons
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Weak interactions and disorder
Analytical, general picture:
Interactions \(\Rightarrow\) transition between weak and strong disorder
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
disorder
interactions
Anderson localized
Delocalized
Ergodic
Insulator
L. Fleischman, P. W. Anderson, PRB 2, 2336 (1980) \(\rightarrow\) single-particle excitations and conditions for Anderson transition
B. Altschuler, Y. Gefen, A. Kamenev, L. S. Levitov, PRL 78, 2803, (1997) \(\rightarrow\) quasi particle lifetime & localization in Fock space
P. Jacquod, D. L. Shepelyansky, PRL 79, 1837 (1997) \(\rightarrow\) Gap ratio statistics, finite systems
I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, PRL 95, 206603 (2005) \(\rightarrow\) zero conductivity at low temperature
*D. M. Basko, I. L. Aleiner, B. L. Altschuler, Annals of Physics 321, 1126 (2006) \(\rightarrow\) metal-insulator transition, localization in Fock space
I.L. Aleiner, B. L. Altshuler, G. V Shlyapnikov, Nature Physics 6, 900-904 (2010) \(\rightarrow\) weakly interacting bosons
17
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
18
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019) F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
A. Pal, D. Huse, PRB 82, 174411 2010
(See series of works by V. Oganesyan, A. Pal, D. Huse, 2007-2010)
Exact diagonalization
disorder
gap ratio
Probes: gap ratio
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
18
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019) F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
Gaussian orthogonal ensemble statistics = random matrix
level repulsion
\(\leftrightarrow\) ergodic
A. Pal, D. Huse, PRB 82, 174411 2010
(See series of works by V. Oganesyan, A. Pal, D. Huse, 2007-2010)
Exact diagonalization
disorder
gap ratio
Probes: gap ratio
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
18
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019) F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
Gaussian orthogonal ensemble statistics = random matrix
level repulsion
\(\leftrightarrow\) ergodic
Poisson statistics
non-ergodic
A. Pal, D. Huse, PRB 82, 174411 2010
(See series of works by V. Oganesyan, A. Pal, D. Huse, 2007-2010)
Exact diagonalization
disorder
gap ratio
Probes: gap ratio
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
19
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019) F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
Probes: gap ratio | entanglement entropy
Initial \(S^z\) basis random product state
+
TEBD
W = 5
Anderson
No growth
of entanglement
J. H. Bardarson, F. Pollmann, and J. E. Moore, PRL 109, 017202 (2012)
M. Znidaric, T. Prosen, and P. Prelovsek PRB 77, 064426 (2008)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019) F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
Probes: gap ratio | entanglement entropy
Anderson
No growth
of entanglement
J. H. Bardarson, F. Pollmann, and J. E. Moore, PRL 109, 017202 (2012)
M. Znidaric, T. Prosen, and P. Prelovsek PRB 77, 064426 (2008)
Many body
Log growth
of entanglement
19
Initial \(S^z\) basis random product state
+
TEBD
W = 5
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
20
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\(\frac{E -E_{\min}}{E_{\max}-E_{\min}}\)
D. J. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
A few among many...
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019) F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
Probes: gap ratio | entanglement entropy | participation entropy
|\Psi\rangle = \sum_{\alpha = 1}^{\mathcal{N}} \psi_{\alpha} |\alpha \rangle
S_q = \frac{1}{1-q} \ln\left(\sum_{\alpha=1}^{\mathcal{N}} |\psi_{\alpha}|^{2q}\right)
Configuration space
S_q = a_q \ln(\mathcal{N})
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
20
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019) F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
Probes: gap ratio | entanglement entropy | participation entropy | imbalance [...]
M. Schreiber et al. (I. Bloch) , Science 349, 842 (2015)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Aubry-André Model with interactions :
1D system of ultracold fermions
W
Ergodic
MBL
2016
Finite L
debate
21
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
W
Ergodic
MBL
2016
Finite L
- Finite-size scaling? Location of the transition?
- Destabilization by ergodic bubbles even at strong disorder?
- Immediate onset of quantum chaos? Intermediate phase(s)?
debate
21
W
Ergodic
MBL phase/ regimes?
Prethermal
regime?
2023
J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, PRE 102, 062144 (2020); D.A. Abanin, et al, Annals of Physics 427, 168415, (2021);
D. Sels, A. Polkovnikov, JCCM January 2023_1 (2023); Tyler LeBlond, Dries Sels, Anatoli Polkovnikov, and Marcos Rigol, PRB 104, L201117 (2021); A. Morningstar et al, PRB 105, 174205 (2022); L. Colmenarez, D. Luitz, W. De Roeck, arXiv:2308.01350 (2023); P, Sierant and J. Zakrzewski, PRB 105, 224203 (2022)...
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
W
Ergodic
MBL
2016
Finite L
- Finite-size scaling? Location of the transition?
- Destabilization by ergodic bubbles even at strong disorder?
- Immediate onset of quantum chaos? Intermediate phase(s)?
debate
21
W
Ergodic
MBL phase/ regimes?
Prethermal
regime?
For today : Magnetization, ED data and comparison to the Anderson line
J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, PRE 102, 062144 (2020); D.A. Abanin, et al, Annals of Physics 427, 168415, (2021);
D. Sels, A. Polkovnikov, JCCM January 2023_1 (2023); Tyler LeBlond, Dries Sels, Anatoli Polkovnikov, and Marcos Rigol, PRB 104, L201117 (2021); A. Morningstar et al, PRB 105, 174205 (2022); L. Colmenarez, D. Luitz, W. De Roeck, arXiv:2308.01350 (2023); P, Sierant and J. Zakrzewski, PRB 105, 224203 (2022)...
2023
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Exact diagonalization
challenge
22
- Many-body
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
challenge
- Many-body
- Disorder \(\rightarrow\) translation invariance, high number of realisations
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
22
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
challenge
22
- Many-body
- Disorder \(\rightarrow\) translation invariance, high number of realisations
- High-energy eigenstates
- High density of eigenstates
- Potential absence of thermalization
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
23
spectral transformation
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
spectral transformation
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
spectral transformation
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
spectral transformation
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
spectral transformation
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
Do not invert!
Solve \((H-\sigma) \vec{y} = \vec{x}\)
23
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
spectral transformation
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
G = "\sum_k \alpha_k H^k"
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
spectral transformation
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
G = "\sum_k \alpha_k H^k"
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
Up to 22, 24 sites
\(\mathcal{N} > 2\cdot 10^6\)
# non-zero el. \(> 3 \cdot 10^7 \)
23
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Magnetization distributions
24
Anderson chain / XX chain
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Magnetization distributions
Anderson chain / XX chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
24
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Magnetization distributions
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
24
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Magnetization distributions
\(L\) increases
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
24
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Magnetization distributions
\(L\) increases
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
24
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Magnetization distributions
25
\(L\) increases
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Minimal deviations in the XX chain - TOy model
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Spin Freezing
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
Spin Freezing
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
Spin Freezing
J. C., N. Laflorencie, arXiv:2305.10574
SPIN FREEZING!
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
SPIN FREEZING!
Some eigenstate
Spin Freezing
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
Toy model : analytical description
27
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
Toy model : analytical description
27
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
Toy model : analytical description
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
27
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Minimal deviation?
28
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle
\ell_{\mathrm{cluster}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
28
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
\ell_{\mathrm{cluster}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
28
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
\ell_{\mathrm{cluster}}
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
28
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
\ell_{\mathrm{cluster}}
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
28
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
:
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
28
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
:
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min}= e^{\overline{\ln\delta_{\min}}} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
28
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\ell_{\mathrm{cluster}}
29
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
29
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
29
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
r
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
29
\gamma
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Exponent : toy model
JC, N. Laflorencie, arXiv:2305.10574
30
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Exponent : toy model
JC, N. Laflorencie, arXiv:2305.10574
\xi = \frac{1}{\ln\left(1+\left(\frac{W}{W_0}\right)^2 \right)}
30
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Exponent : toy model
JC, N. Laflorencie, arXiv:2305.10574
\xi = \frac{1}{\ln\left(1+\left(\frac{W}{W_0}\right)^2 \right)}
30
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Chain breaking
JC, N. Laflorencie, arXiv:2305.10574
31
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Chain breaking
JC, N. Laflorencie, arXiv:2305.10574
31
\delta^{\mathrm{typ}}_{\min} = e^{\overline{\ln\delta_{\min}}} \approx L^{-\gamma_{\mathrm{typ}}(W)}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Chain breaking
JC, N. Laflorencie, arXiv:2305.10574
31
\delta^{\mathrm{typ}}_{\min} = e^{\overline{\ln\delta_{\min}}} \approx L^{-\gamma_{\mathrm{typ}}(W)}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Exponents: XX CHain
32
JC, N. Laflorencie, arXiv:2305.10574
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
JC, N. Laflorencie, arXiv:2305.10574
ED
1/\gamma_{\mathrm{typ}}
Exponents: XX CHain
32
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
JC, N. Laflorencie, arXiv:2305.10574
Agreement
ED - Toy model !
ED
1/\gamma_{\mathrm{typ}}
Exponents: XX CHain
32
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Scaling
33
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} ?
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Scaling
JC, N. Laflorencie, arXiv:2305.10574
\(L \gg \xi\)
33
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} ?
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Scaling
JC, N. Laflorencie, arXiv:2305.10574
33
\(L \gg \xi\)
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} ?
- \frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi} -c
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Quantitative description : Extreme value statistics
Tails and extremes
34
Tails
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tails and extremes
34
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
So far :
Extreme value
Tails
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tails and extremes
34
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
So far :
Extreme value
Tails
Extreme value theory
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Z = \mathrm{rescaled}(Y)
Extreme value theory
Extreme value
Tails
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Power-law tail
Z = \mathrm{rescaled}(Y)
Extreme value theory
Extreme value
Tails
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Power-law tail
Z = \mathrm{rescaled}(Y)
z \sim \frac{\beta}{z^{1+\beta}} e^{-z^{-\beta}}
Fréchet law
Extreme value theory
Extreme value
Tails
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Power-law tail
Exponential tail
Gaussian tail
Z = \mathrm{rescaled}(Y)
z \sim \frac{\beta}{z^{1+\beta}} e^{-z^{-\beta}}
Fréchet law
Extreme value theory
Extreme value
Tails
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Power-law tail
Exponential tail
Gaussian tail
z \sim e^{-z - e^{-z}}
Gumbel law
Z = \mathrm{rescaled}(Y)
z \sim \frac{\beta}{z^{1+\beta}} e^{-z^{-\beta}}
Fréchet law
Extreme value theory
Extreme value
Tails
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
36
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
36
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
37
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
37
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
37
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
37
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
EXPONENT : POWER-LAW
JC, N. Laflorencie, arXiv:2305.10574
38
ED
1/\gamma_{\mathrm{typ}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
JC, N. Laflorencie, arXiv:2305.10574
1+\alpha
ED
1/\gamma_{\mathrm{typ}}
EXPONENT : POWER-LAW
38
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Consequences : interacting system
39
Effect of interactions?
\mathcal{H} = \mathcal{H}_{XX} + \Delta \sum_i S_i^z S_{i+1}^z
"Stability" of the cluster with respect to the interactions?
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
40
Effect of interactions?
disorder
increases
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
40
Effect of interactions?
disorder
increases
\(\delta_i\) \(\rightarrow\) 1/2 (\(\langle S^z_i\rangle \rightarrow 0\))
Empty circles:
Heisenberg
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
40
Effect of interactions?
disorder
increases
Empty circles:
Heisenberg
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\(\delta_i\) \(\rightarrow\) 1/2 (\(\langle S^z_i\rangle \rightarrow 0\))
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
41
Exponent
At strong disorder, \(\gamma \sim 1/\xi\)
\gamma
\(\Rightarrow \) Interpretation of the exponent as related to a many-body localization length
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
42
Exponent
At strong disorder, \(\gamma \sim 1/\xi\)
\gamma
\(\Rightarrow \) Interpretation of the exponent as related to a many-body localization length
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\(\Lambda\) : disorder-dependent non-ergodicity volume
\(\lambda\) : interpreted as a localization length
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
43
Extreme value distributions
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
JC, N. Laflorencie, arXiv:2305.10574
\(W = 2\)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
43
Extreme value distributions
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
JC, N. Laflorencie, arXiv:2305.10574
\(W = 2\)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
43
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
\(\Delta = 0\)
\(\Delta = 1\)
Extreme value distributions
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
43
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
\(\Delta = 0\)
\(\Delta = 1\)
Extreme value distributions
JC, N. Laflorencie, arXiv:2305.10574
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
44
Kullback-Leibler divergence
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
44
Kullback-Leibler divergence
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
JC, N. Laflorencie, arXiv:2305.10574
E. H. V. Doggen et al., PRB 98, 174202 (2018)
See e.g. D. Sels, PRB 106 , L020202 (2022)
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
44
Consequences : Heisenberg
E. H. V. Doggen et al., PRB 98, 174202 (2018)
See e.g. D. Sels, PRB 106 , L020202 (2022)
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
JC, N. Laflorencie, arXiv:2305.10574
- Transition in the extreme value distributions
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
44
Consequences : Heisenberg
- Transition in the extreme value distributions
- Coinciding with the MBL transition?
E. H. V. Doggen et al., PRB 98, 174202 (2018)
See e.g. D. Sels, PRB 106 , L020202 (2022)
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
JC, N. Laflorencie, arXiv:2305.10574
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Take home message
SPIN FREEZING!
45
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
45
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
- Excellent fits & collapses with a Fréchet Law
45
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
Heisenberg chain at strong disorder:
Chain breaks!
- Excellent fits & collapses with a Fréchet Law
45
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
Heisenberg chain at strong disorder:
Chain breaks!
- Excellent fits & collapses with a Fréchet Law
45
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Comparing \(\delta_{\min}\) deviation in Heisenberg vs Many-body Anderson:
Extreme value transition characterized by the
KL divergence.
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
Heisenberg chain at strong disorder:
Chain breaks!
- Excellent fits & collapses with a Fréchet Law
Comparing \(\delta_{\min}\) deviation in Heisenberg vs Many-body Anderson:
Extreme value transition characterized by the
KL divergence.
Thank you for your attention!
45
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Thank you for the question!
Anderson Localization : example
1 particle
11
Billy J, et al. Direct observation of Anderson localization of matter waves in a controlled disorder. Nature. 2008
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
AL vs MBL
No spreading of entanglement
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Weak interactions and disorder
Analytical, general picture:
Interactions \(\Rightarrow\) transition between weak and strong disorder
L. Fleischman, P. W. Anderson, PRB 2, 2336 (1980) \(\rightarrow\) single-particle excitations and conditions for Anderson transition
B. Altschuler, Y. Gefen, A. Kamenev, L. S. Levitov, PRL 78, 2803, (1997) \(\rightarrow\) quasi particle lifetime & localization in Fock space
P. Jacquod, D. L. Shepelyansky, PRL 79, 1837 (1997) \(\rightarrow\) Gap ratio statistics, finite systems
I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, PRL 95, 206603 (2005) \(\rightarrow\) zero conductivity at low temperature
*D. M. Basko, I. L. Aleiner, B. L. Altschuler, Annals of Physics 321, 1126 (2006) \(\rightarrow\) metal-insulator transition, localization in Fock space
I.L. Aleiner, B. L. Altshuler, G. V Shlyapnikov, Nature Physics 6, 900-904 (2010) \(\rightarrow\) weakly interacting bosons
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
disorder
interactions
Anderson localized
Delocalized
Ergodic
Insulator
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
ED : Why
High energy eigenstates, high dos
Potential absence of thermalization
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
Cannot use stochastic methods
Usual condensed matter methods target the ground state.
\(\rightarrow\) DMRG-X, RSRG-X, time evolution with MPS, Unitary flow, ...
Ideally should work on both sides of the transition
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Eigenstate thermalization hypothesis
Fate of isolated many-body quantum systems?
\(\leftrightarrow\) A question from quantum chaos:
Thermal average
Time average
?
ETH
J. M. Deutsch , PRA. 43, 2046–2049, (1991) , M. Srednicki, PRE 50, 888–901, (1994)
L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Eigenstate thermalization hypothesis
Fate of isolated many-body quantum systems?
\(\leftrightarrow\) A question from quantum chaos:
Thermal average
Time average
?
ETH
J. M. Deutsch , PRA. 43, 2046–2049, (1991) , M. Srednicki, PRE 50, 888–901, (1994)
L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016)
(i) diagonal elements of local observables are smooth functions of the energy and take their microcanonical average value
(ii)off-diagonal elements vanish in the thermodynamic limit like \(e^{(-E_m -E_n)/2}\)
Statement about high energy.
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Heisenberg: Many-Body Localization
Morningstar et al, PRB 105, 174205 (2022)
Fate of isolated quantum many-body systems ?
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Kane-fisher
LL + weakened link can flow to
an opened chain.
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Entanglement entropy
N. Laflorencie, in A. Bayat et al. (eds.), Entanglement in Spin Chains,
Quantum Science and Technology, https://doi.org/10.1007/978-3-031-03998-0_4
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Distributions of minimal deviations
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Distributions of minimal deviations
JC, Laflorencie, In preparation
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Cluster lengths
JC, Laflorencie, In preparation
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Power-law tails?
JC, N. Laflorencie, arXiv:2305.10574
\delta_{\min} \sim e^{-\frac{\ell}{2\xi}}
\( \Rightarrow \delta\) occurs if there is \(\ell \geq -2 \xi \ln(\delta)\)
\ell
\mathcal{P}(\ell) \propto 2^{-\ell}
Very roughly:
\Rightarrow \mathcal{P}_L(\ln(\delta)) \propto \exp\left(2\xi\ln2 \times \ln(\delta)\right)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Power-law tails?
JC, N. Laflorencie, arXiv:2305.10574
\delta_{\min} \sim e^{-\frac{\ell}{2\xi}}
\( \Rightarrow \delta\) occurs if there is \(\ell \geq -2 \xi \ln(\delta)\)
\ell
\mathcal{P}(\ell) \propto 2^{-\ell}
Very roughly:
\mathcal{P}_L(\delta) \propto \delta^{\left(2\xi\ln2-1\right)}
\Rightarrow \mathcal{P}_L(\ln(\delta)) \propto \exp\left(2\xi\ln2 \times \ln(\delta)\right)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Experiments
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
PROBES
"Dynamical" probes
"Static" probes
Entanglement
entropy (EE)
Multifractality (participation entropy)
Entanglement spectrum (ES)
and these probes -->
Spectral repulsion
Magnetization & extremal magnetization
-> Lcluster, delta min
Evolution / autocorrelation of a prepared state
Out-of-Time-Order Correlator (OTOC)
KL divergence between eigenstates
Increasing number of involved states
Gap ratio
two-eigenstates correlation functions
Spectral form factor
Level compressibility
Imbalance
Dream: LIOMs
(minimal correlator)
Many-body resonances between eigenstates
Minimal gap
Distribution of matrix elements?
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Heisenberg: Ergodic to MBL
Fate of isolated quantum many-body systems ?
\(\frac{E -E_{\min}}{E_{\max}-E_{\min}}\)
D. J. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
\(S_E\) : Entanglement entropy (red: visual estimate)
\(r\) : Gap ratio
\(\mathcal{F}\) : Bipartite fluctuations
\(\mathcal{F} = \langle (S_A^z)^2 \rangle - \langle S_A^z \rangle ^2 \)
\(S_1^{P} = a_1 \ln(\mathrm{dim} H) = - \sum_i p_i \ln(p_i)\) : partitipation entropy (multifractality)
\( f \) : dynamic fraction of an
initial spin polarization
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Hilbert-space localization
|\Psi\rangle = \sum_{\alpha = 1}^{\mathcal{N}} \psi_{\alpha} |\alpha \rangle
S_q = \frac{1}{1-q} \ln\left(\sum_{\alpha=1}^{\mathcal{N}} |\psi_{\alpha}|^{2q}\right)
\mathcal{N} = \text{ Hilbert space dim.}
Participation entropies
Basis-dependent
Macé, Alet, Laflorencie, PRL 123, 180601 (2019)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
POLFED
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Ground state phase diagram
\Delta
h
\Delta = 0
\Delta = -1
Bose glass:
gapless, but exponentially decaying correlations
finite compressibility
insulating; localized
infinite superfluid susceptibility
\Delta = -1/2
BKT from SDRG:
K > 3/2
Weak link physics
Superfluid
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Dynamic fraction
D. J. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
J. COLBOIS | EVT AND LOCALIZATION | LPMMC | 07.11.2023
Extreme Statistics in Random spin chains
By Jeanne Colbois
Extreme Statistics in Random spin chains
LPMMC seminar
